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Hansen, M. T., Romero-Lopez, F., & Sharpe, S. R. (2020). Generalizing the relativistic quantization condition to include all three-pion isospin channels. J. High Energy Phys., 07(7), 047–49pp.
Abstract: We present a generalization of the relativistic, finite-volume, three-particle quantization condition for non-identical pions in isosymmetric QCD. The resulting formalism allows one to use discrete finite-volume energies, determined using lattice QCD, to constrain scattering amplitudes for all possible values of two- and three-pion isospin. As for the case of identical pions considered previously, the result splits into two steps: the first defines a non-perturbative function with roots equal to the allowed energies, E-n(L), in a given cubic volume with side-length L. This function depends on an intermediate three-body quantity, denoted K-df;3, which can thus be constrained from lattice QCD input. The second step is a set of integral equations relating K-df,K-3 to the physical scattering amplitude, M-3. Both of the key relations, E-n(L) <-> K-df,K-3 and K-df,K-3 <-> M-3, are shown to be block-diagonal in the basis of definite three-pion isospin, I-pi pi pi, so that one in fact recovers four independent relations, corresponding to I-pi pi pi = 0; 1; 2; 3. We also provide the generalized threshold expansion of K-df,K-3 for all channels, as well as parameterizations for all three-pion resonances present for I-pi pi pi = 0 and I-pi pi pi = 1. As an example of the utility of the generalized formalism, we present a toy implementation of the quantization condition for I-pi pi pi = 0, focusing on the quantum numbers of the omega and h(1) resonances.
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Donini, A., Hernandez, P., Pena, C., & Romero-Lopez, F. (2020). Dissecting the Delta I=1/2 rule at large N-c. Eur. Phys. J. C, 80(7), 638–12pp.
Abstract: We study the scaling of kaon decay amplitudes with the number of colours, N-c, in a theory with four degenerate flavours, N-f = 4. In this scenario, two current-current operators, Q(+/-), mediate Delta S = 1 transitions, such as the two isospin amplitudes of non-leptonic kaon decays for K -> (pi pi)(I=0,2), A(0) and A(2.) In particular, we concentrate on the simpler K -> pi amplitudes, A(+/-), mediated by these two operators. A diagrammatic analysis of the large-N-c scaling of these observables is presented, which demonstrates the anticorrelation of the leading O(1/N-c) and O(N-f/N-c(2)) corrections in both amplitudes. Using our new N-f = 4 and previous quenched data, we confirm this expectation and show that these corrections are naturally large and may be at the origin of the Delta I = 1/2 rule. The evidence for the latter is indirect, based on the matching of the amplitudes to their prediction in Chiral Perturbation Theory, from which the LO low-energy couplings of the chiral weak Hamiltonian, g(+/-), can be determined. A NLO estimate of the K -> (pi pi)(I=0,2) isospin amplitudes can then be derived, which is in good agreement with the experimental value.
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Blanton, T. D., Romero-Lopez, F., & Sharpe, S. R. (2020). I=3 Three-Pion Scattering Amplitude from Lattice QCD. Phys. Rev. Lett., 124(3), 032001–7pp.
Abstract: We analyze the spectrum of two- and three-pion states of maximal isospin obtained recently for isosymmetric QCD with pion mass M approximate to 200 MeV in Horz and Hanlon, [Phys. Rev. Lett. 123, 142002 (2019)]. Using the relativistic three-particle quantization condition, we find similar to 2 sigma evidence for a nonzero value for the contact part of the 3 pi(+) (I = 3) scattering amplitude. We also compare our results to leading-order chiral perturbation theory. We find good agreement at threshold and some tension in the energy dependent part of the 3 pi(+) scattering amplitude. We also find that the 2 pi(+) (I = 2) spectrum is fit well by an s-wave phase shift that incorporates the expected Adler zero.
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Romero-Lopez, F., Sharpe, S. R., Blanton, T. D., Briceno, R. A., & Hansen, M. T. (2019). Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states. J. High Energy Phys., 10(10), 007–43pp.
Abstract: In this work, we use an extension of the quantization condition, given in ref. [1], to numerically explore the finite-volume spectrum of three relativistic particles, in the case that two-particle subsets are either resonant or bound. The original form of the relativistic three-particle quantization condition was derived under a technical assumption on the two-particle K matrix that required the absence of two-particle bound states or narrow two-particle resonances. Here we describe how this restriction can be lifted in a simple way using the freedom in the definition of the K-matrix-like quantity that enters the quantization condition. With this in hand, we extend previous numerical studies of the quantization condition to explore the finite-volume signature for a variety of two- and three-particle interactions. We determine the spectrum for parameters such that the system contains both dimers (two-particle bound states) and one or more trimers (in which all three particles are bound), and also for cases where the two-particle subchannel is resonant. We also show how the quantization condition provides a tool for determining infinite-volume dimer-particle scattering amplitudes for energies below the dimer breakup. We illustrate this for a series of examples, including one that parallels physical deuteron-nucleon scattering. All calculations presented here are restricted to the case of three identical scalar particles.
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Hernandez, P., Pena, C., & Romero-Lopez, F. (2019). Large Nc scaling of meson masses and decay constants. Eur. Phys. J. C, 79(10), 865–13pp.
Abstract: We perform an ab initio calculation of the Nc scaling of the low-energy couplings of the chiral Lagrangian of low-energy strong interactions, extracted from the mass dependence of meson masses and decay constants. We compute these observables on the lattice with four degenerate fermions, Nf=4, and varying number of colours, Nc=3-6, at a lattice spacing of a similar or equal to 0.075 fm. We find good agreement with the expected Nc scaling and measure the coefficients of the leading and subleading terms in the large Nc expansion. From the subleading Nc corrections, we can also infer the Nf dependence, that we use to extract the value of the low-energy couplings for different values of Nf. We find agreement with previous determinations at Nc=3 and Nf=2,3 and also, our results support a strong paramagnetic suppression of the chiral condensate in moving from Nf=2 to Nf=3.
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